A cartel could hold prices high, or settle on a dozen other arrangements, and each one, once everyone expects it, sustains itself. Repeated play does not predict cooperation so much as it predicts almost anything.
Folk theorems
The last lesson showed cooperation can survive if players are patient. The folk theorem says far more, and it is a warning as much as a result: when the future matters enough, nearly every outcome that clears a low bar can be held up as an equilibrium. Repetition explains too much.
In the repeated PD, cooperation survives under grim trigger when delta is...
In a strictly competitive game, the most a player can guarantee themselves alone is called the...
Predict, then play
The plot is the space of average payoffs in the repeated Prisoner’s Dilemma. The teal region is what the game can produce; the shaded box is where both players clear their minmax of 1. Drag a target and see whether it can be sustained.
Why did that happen?
Take a guess, then reveal the reason
The engine is the threat. Any deviation can be met with punishment that drives the deviator down to their minmax, the lowest payoff the others can hold them to when they best-respond. This differs from maximin, the highest payoff a player can guarantee without controlling the others. As long as the target leaves both players above the relevant minmax level, future punishment can outweigh the one-time gain from cheating.
The math
The feasible payoffs are the convex hull of the stage-game outcomes: for the PD with , the hull of . A player’s minmax is the lowest payoff the others can force on them while the player best-responds; in the PD that is (defect against a defector).
For an infinitely repeated game, take any feasible payoff profile in which every player earns strictly more than their minmax value. Then for discount factors close enough to 1, there is a subgame-perfect equilibrium of the repeated game whose average payoff is exactly that profile. This guarantees the strict interior of the feasible, individually rational region. Boundary points need separate analysis: some are equilibria too, but the strict theorem does not settle them. (With two players that is the whole statement; with three or more, Fudenberg and Maskin also need the payoff set to be full-dimensional, so that a deviator can be punished without punishing the punishers.)
So is an equilibrium, but so is , and so are countless asymmetric splits. Two flavors: the Nash-threat version (Friedman) punishes with a stage-game Nash equilibrium; the stronger minmax version (Fudenberg-Maskin) punishes down to the minmax, which enlarges the supportable set.
The lesson is sobering. Repetition does not pick out cooperation; it makes prediction hard, because so many outcomes are consistent with equilibrium.
Work it out
Support a point, and check the floor
Symmetric. To hold , both play grim trigger: cooperate, and defect forever if the other ever does. A one-time defection gains now but forfeits the cooperative stream, so for high enough nobody deviates. Standard cooperation.
Asymmetric. To hold , alternate outcomes over time (say, favouring player 1 in some rounds) so the long-run averages land there, and back it with the threat of minmax punishment. Both averages exceed the minmax of 1, so each player prefers the arrangement to being punished, and the profile is supportable.
The floor. Try : both are below the minmax of 1. A player can guarantee 1 by simply defecting forever, so no equilibrium can pin them at 0.5. The individually rational bound is what rules a target out.
Key idea
With patient players, every feasible payoff above the minmax is an equilibrium of the repeated game. The folk theorem does not predict cooperation; it shows repetition can rationalize almost anything.
Check yourself
Sources
- Fudenberg, D., and Maskin, E. (1986). The folk theorem in repeated games with discounting or with incomplete information. Econometrica, 54(3), 533-554.
- Friedman, J. W. (1971). A non-cooperative equilibrium for supergames. Review of Economic Studies, 38(1), 1-12.